Positive definite function zoo

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$.

For a definition and discussion of positive definite function see here.

I've often wished I had a collection of diverse examples of positive definite functions on groups, for the purpose of testing various conjectures. I hope the diverse experience of the participants of this forum can help me collect a list of such examples.

To clarify what I'd like to see:

What is an example of a positive definite function on a group $G$ that is not easily seen to be a coefficient of a unitary representation of $G$? What are some positive definite functions that arise in contexts sufficiently removed from studying the coefficients of unitary representations?

Also, the weirder the group $G$ the better.

Edit: There is now a version of this question on MO.

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As per request, I've made the question CW. –  Zev Chonoles Aug 13 '11 at 23:42
Thank you, Zev! –  Jon Bannon Aug 13 '11 at 23:45
I guess my favourite example is $\frac{1}{1+x^2}$ on $\mathbb{R}$. But it's not quite clear what exactly you're looking for: explicit positive elements of the Fourier algebra or would a function of the form $f \ast \tilde{f}$ for $f \in L^2(G)$ already be satisfactory for you? I find the exposition on positive definite functions in appendix C of Bekka-de la Harpe-Valette quite nice, but sect 13.4 and the following chapters of Dixmier's book on $C^\ast$-algebras is still the best source in my opinion (few explicit examples, though). –  t.b. Aug 14 '11 at 10:06
let me clarify, Theo. –  Jon Bannon Aug 14 '11 at 11:43
Thanks for the tag edits, someone! –  Jon Bannon Aug 14 '11 at 11:54